Pedigree-free animal models: the relatedness
Francesca D. Frentiu1,2,3, Sonya M. Clegg4, John Chittock3, Terry Burke3,
Mark W. Blows1and Ian P. F. Owens4,5,*
1School of Integrative Biology, University of Queensland, St Lucia, Queensland 4072, Australia
2Department of Ecology and Evolutionary Biology, University of California, Irvine, CA 92697, USA
3Department of Animal and Plant Sciences, University of Sheffield, Sheffield S10 2TN, UK
4Division of Biology and5NERC Centre for Population Biology, Imperial College London, Silwood Park,
Ascot, Berkshire SL5 7PY, UK
Animal models typically require a known genetic pedigree to estimate quantitative genetic parameters.
Here we test whether animal models can alternatively be based on estimates of relatedness derived entirely
from molecular marker data. Our case study is the morphology of a wild bird population, for which we
report estimates of the genetic variance–covariance matrices (G) of six morphological traits using three
methods: the traditional animal model; a molecular marker-based approach to estimate heritability based
on Ritland’s pairwise regression method; and a new approach using a molecular genealogy arranged in a
relatedness matrix (R) to replace the pedigree in an animal model. Using the traditional animal model, we
found significant genetic variance for all six traits and positive genetic covariance among traits. The
pairwise regression method did not return reliable estimates of quantitative genetic parameters in this
population, with estimates of genetic variance and covariance typically being very small or negative. In
contrast, we found mixed evidence for the use of the pedigree-free animal model. Similar to the pairwise
regression method, the pedigree-free approach performed poorly when the full-rank R matrix based on the
molecular genealogy was employed. However, performance improved substantially when we reduced the
dimensionality of the R matrix in order to maximize the signal to noise ratio. Using reduced-rank R
matrices generated estimates of genetic variance that were much closer to those from the traditional model.
Nevertheless, this method was less reliable at estimating covariances, which were often estimated to be
negative. Taken together, these results suggest that pedigree-free animal models can recover quantitative
genetic information, although the signal remains relatively weak. It remains to be determined whether this
problem can be overcome by the use of a more powerful battery of molecular markers and improved
methods for reconstructing genealogies.
Keywords: quantitative genetics; animal model; relatedness; molecular markers; morphology; birds
The application of animal models to wild populations
promises to revolutionize our understanding of evolution-
ary genetics in natural environments (Kruuk 2004). This
is because animal models, in their broadest sense, are
simply individual-based mixed models that use a known
pedigree to estimate relatedness among individuals and
thereby estimate a range of quantitative genetic par-
ameters (Lynch & Walsh 1998). The key reason that
animal models offer such promise for the study of wild
populations is that this approach can use a natural
pedigree to extract quantitative genetic information
under natural conditions. In contrast, most quantitative
genetic techniques require breeding experiments and are
agricultural studies (Falconer & Mackay 1996). Animal
models have now been applied to a number of populations
to tackle questions as diverse as the heritability of fitness
(Kruuk et al. 2000), evolutionary stasis (Merila ¨ et al.
2001; Kruuk et al. 2002), sexual selection and coloration
(Hadfield & Owens 2006; Hadfield et al. 2006, 2007),
condition dependence (Gleeson et al. 2005), parental care
(MacColl & Hatchwell 2003), the genetic consequences
of harvesting (Coltman et al. 2003) and the evolutionary
response to climate change (Brommer et al. 2005).
Such widespread interest in the animal model approach
has, however, led to the realization that the need for a
known pedigree is itself a limitation. It is no coincidence
that most studies to date using the animal model concern
populations that have been the subject of long-term
projects (Kruuk et al. 2000, 2001, 2002; Merila ¨ &
Sheldon 2000; Merila ¨ et al. 2001; Coltman et al. 2003;
Garant et al. 2004, 2005; McCleery et al. 2004;
Charmantier et al. 2006a,b). The need for long-term
information on individual patterns of mating and repro-
duction limits the range and type of populations where an
animal model can be used. One way potentially to
overcome this limitation is to use molecular marker data
Proc. R. Soc. B (2008) 275, 639–647
Published online 23 January 2008
Electronic supplementary material is available at http://dx.doi.org/10.
1098/rspb.2007.1032 or via http://journals.royalsociety.org.
One contribution of 18 to a Special Issue ‘Evolutionary dynamics of
*Author and address for correspondence: Division of Biology,
Imperial College London, Silwood Park, Ascot, Berkshire SL5 7PY,
Received 28 August 2007
Accepted 17 October 2007
This journal is q 2008 The Royal Society
to estimate the genetic relationships among individuals in
a population and then use the resulting relatedness matrix,
instead of a known pedigree, to construct the animal
model (Lynch & Walsh 1998; Garant & Kruuk 2005;
Rodrı ´guez-Ramilo et al. 2007). This approach could allow
the animal model framework to be extended to any
population for which it was possible to obtain reliable
estimates of relatedness based on molecular marker data
(Moore & Kukuk 2002), which would greatly expand the
rangeof potential applications if the approach provedtobe
robust. Such an approach has yet to be fully implemented
in any population, however.
The overall aim of this study was therefore to test
whether animal models can indeed be based on estimates
of relatedness derived entirely from molecular marker
data. The idea of estimating quantitative genetic par-
ameters using relatedness estimates derived from molecu-
lar marker data has been explored by a number of workers
(Mousseau et al. 1998; Thomas & Hill 2000; Thomas
et al. 2000; Thomas 2005) and, in particular, has been
developed by Ritland (1996, 2000a,b; Ritland & Ritland
1996). Although Ritland’s method is conceptually similar
to the pedigree-free animal models that we discuss here,
there are key differences between the two. The most
important of these is that Ritland’s method is based on
regressing pairwise estimates of phenotypic similarity on
pairwise estimates of genetic relatedness (Ritland 1996).
Limitations of this approach include difficulties in
estimating significance due to non-independence of
relatedness estimates and that method of moments
relatedness measures do not provide estimates that are
internally consistent across the entire population. In
contrast, the pedigree-free animal model approach we
explore here is based on a relatedness matrix that is
positive definite (i.e. relatedness among multiple individ-
uals are congruous), and allows the full power (and
convenience) of the animal model to be applied when
estimating quantitative genetic parameters.
We apply the pedigree-free animal model to the
example of a free-living population of Capricorn silvereyes
(Zosterops lateralis chlorocephalus) on Heron Island, a small
coral cay on the Australian Great Barrier Reef. This study
population is well suited to our needs because we can
construct a known pedigree from behavioural information
(Kikkawa & Wilson 1983; Robertson et al. 2001), which
we have previously used to estimate heritabilities and
genetic correlations for a series of morphological traits
using a pedigree-based animal model (Frentiu et al. 2007).
In addition, we have a dataset that consists of individuals
used in a cross-fostering experiment (Frentiu et al. 2007),
minimizing the extent to which the effect of shared genes is
inflated by the effect of shared environments. Finally, the
quantitative genetic basisof morphologyin this population
of birds is of intrinsic interest because it is an example of
an unusually large island race that shows the characteristic
pattern for insular passerines (Clegg & Owens 2002),
having evolved to be approximately 40% larger than its
mainland counterpart in just 4000 years (Clegg et al.
2002a,b; Robinson-Wolrath & Owens 2003; Scott et al.
2003; Frentiu et al. 2007).
The specific aims of the study were to: (i) develop a
series of polymorphic molecular markers and determine
whether they were able to differentiate between close
relatives (full siblings) and unrelated individuals,
(ii) estimate quantitative genetic parameters for six
morphological traits using Ritland’s pairwise regression
method and compare these with the estimates from a
pedigree-based traditional animal model, (iii) develop a
pedigree-free animal model using a molecular genealogy,
(iv) determine the effectiveness of the pedigree-free versus
the traditional animal model in estimating quantitative
genetic parameters, and (v) explore methods to increase
the power of the molecular genealogy approach.
2. MATERIAL AND METHODS
(a) Study population and morphological
The individuals used in this study (NZ479) were the same as
those described in a previous study of this population
(Frentiu et al. 2007). Briefly, measurements and samples
were taken from nestlings in the Heron Island population of
Capricorn silvereyes over three consecutive breeding seasons:
2000–2001, 2001–2002 and 2003–2004. The modal clutch
size in this population is three, so our expectation was that the
sample would contain a substantial number of full siblings.
The presence of such full siblings was anticipated to be
helpful because it would allow us to check the power of our
molecular markers in identifying close relatives. As part of a
long-term study of evolutionary genetics in this population,
we also conducted reciprocal cross-foster manipulations in
which one or two chicks were swapped between pairs of nests
and this design was used to implement the traditional animal
model (see Frentiu et al. 2007).
Morphological measurements were taken as described by
Frentiu et al. (2007). Briefly, six morphological measure-
ments were obtained: wing length; tail length; tarsus length;
culmen (bill) length; culmen width; and culmen depth. All
measurements were standardized to unit variance and a mean
of zero before analysis.
(b) Molecular markers and genotyping
Genomic DNA was extracted from blood samples using a
modified salting-out protocol (Nicholls et al. 2000). Eleven
microsatellite markers were available from previous studies
(Degnan et al. 1999; Frentiu et al. 2003) or obtained by
screening other passerine primers for amplification in the
silvereye (for details see Frentiu 2004). Loci were tested for
deviations from Hardy–Weinberg equilibrium, linkage dis-
equilibrium, selective neutrality and sex linkage using the
software MSA (Dieringer & Schlo ¨tterer 2003). There was
no evidence of null alleles, which can skew relatedness
estimates, as indicated by patterns of inheritance across
eight silvereye families and by checking the entire dataset
using the software MICROCHECKER (Van Oosterhout et al.
Individuals were genotyped at the selected 11 loci using
multiplex PCR in 10 ml volumes containing 15 ng of DNA,
1.0 mM of each primer, 0.2 mM of each dNTP and 0.25 units
DNA polymerase (ThermoprimePlus, ABgene, UK) in the
manufacturer’s buffer, including 1.0 or 1.5 mM MgCl2. PCR
cycling consisted of 948C for 3 min, followed by 33 cycles of
45 s at 948C, 45 s at the appropriate annealing temperature
(Frentiu et al. 2003) and 20 s at 728C. One primer in each
pair was labelled with one of three fluorescent dyes (Applied
Biosystems, Warrington, UK). PCR products were analysed
using an ABI 377 DNA sequencer and the GENOTYPER
Software (Applied Biosystems, Warrington, UK).
640F. D. Frentiu et al.Pedigree-free animal models
Proc. R. Soc. B (2008)
Finally, as a measure of the informativeness of the
markers, we calculated exclusion probabilities for parentage
assignment, defined as the ability to exclude a ‘random’
individual from parentage when the other parent’s genotype is
known (Jamieson & Taylor 1997), for each individual marker
and then all markers combined.
(c) Marker-based estimates of relatedness
In order to test the performance of both our markers and
various methods of estimating relatedness, we estimated
relatedness for pairs of individuals expected to be ‘unrelated’
and those expected to be full siblings. Pairs of full siblings
were initially inferred on the basis of behavioural observations
and were subsequently confirmed using the program COLONY
(Wang 2004a,b). Relatedness was inferred from molecular
markers using three widely used pairwise methods: Queller &
Goodnight (QR; 1989); Lynch & Ritland (LR; 1999); and
Wang (W; 2002) using allele frequencies based on all typed
individuals using the program SPAGEDI v. 1.2 (Hardy &
Vekemans 2002). We also estimated relatedness for both
categories of relationship using an alternative algorithm based
on the simulated annealing approach of Ferna ´ndez & Toro
(FT; 2006). We used full siblings and unrelated individuals as
categories because these comprise the two main types of
relationship categories present in this dataset.
In addition, because it has been noted that variance in
relatedness in a population sample will have a substantial
effect on the power to calculate quantitative genetic
parameters (Ritland & Ritland 1996; Ritland 2000b;
Garant & Kruuk 2005; Csille ´ry et al. 2006), we calculated
the actual variance of relatedness, var(rij) (Ritland & Ritland
1996), for each estimator in the program SPAGEDI v. 1.2
(Hardy & Vekemans 2002). Standard errors of var(rij) were
calculated by bootstrapping across loci.
(d) Estimating quantitative genetic parameters
(i) Traditional animal model
For the traditional animal model based on known pedigree
information, we followed the methods described in Frentiu
et al. (2007). Briefly, additive genetic variances and
covariances were estimated from nestling data for all six
morphological traits using a restricted maximum-likelihood
(REML) animal model (Kruuk 2004), using ASREML v. 1.0
(Gilmour et al. 2002). This model included year of sampling,
hatching date and their interaction as fixed effects, while nest
of rearing was included as an additional random effect. The
known pedigree was based on social interactions alone
because a previous study has shown that extra-pair paternity
is absent in this population (Robertson et al. 2001).
(ii) Ritland’s pairwise regression method
For the pairwise regression method, we used the program
MARK v. 3.0 (Ritland 2004) to calculate heritabilities and
genetic correlations for the same six morphological traits
based on nestling data. Thus, no pedigree information was
used in this model. The Queller & Goodnight (1989)
estimator of relatedness was used because, according to the
method of Wang (2006), this estimator was the most
appropriate to use in estimating relatedness (KININFOR
software: Wang 2006). We used spectral decomposition to
test whether the pairwise relatedness estimates would
produce a positive-definite matrix, as indicated by all
eigenvalues being positive. It is important to note that,
owing to the constraints of the pairwise regression technique,
no additional effects were fitted to this model, whereas for the
other two types of model we were able to include year of
measurement, hatching date and nest of rearing. Never-
theless, this lack of additional factors fits with the rationale of
the Ritland method, which aims to estimate quantitative
genetic parameters from marker-based relatedness estimates,
even in the absence of other types of information (Ritland
(iii) Molecular genealogy method
There are at least two problems in attempting to apply a
pedigree-free animal model based on a relatedness matrix
inferred from molecular data alone. First, relatedness
estimates need to be represented in a positive-definite
symmetrical relatedness matrix, R, to take the place of the
pedigree-based relatedness matrix used in the traditional
animal model. Although pairwise estimates of relatedness
could be arranged in such a fashion, the resulting relatedness
matrix will not be positive definite due to internal incon-
sistencies in relatedness among multiple individuals gener-
ated by pairwise estimators. Second, pedigrees in animal
models are assumed to be known without error, and although
there has been some recent work on the effect of misassigned
paternities on heritability estimates (Charmantier & Re ´ale
2005), it is unclear in the context of an animal model how
error will influence the performance of the mixed model
(Lynch & Walsh 1998).
To overcome these problems, we implemented the
simulated annealing approach to the estimation of relatedness
developed by Ferna ´ndez & Toro (2006) to generate a
genealogy based on the molecular marker data, rather than
a relatedness matrix per se. The key properties of a genealogy
in this context are that, unlike relatedness matrices based on
pairwise estimates, all relationships between pairs of
individuals are symmetrical and all relationships across
individuals are consistent with one another. The software
MOL_COAN (Ferna ´ndez & Toro 2006) was used to obtain
this genealogy, with the following control parameters for the
simulated annealing procedure: maximum number of
steps allowedZ300; number of solutions tested at each
stepZ5000; initial temperatureZ0.01; and rate of increase of
temperature Z0.09. In addition, we specified the numbers of
generations present as three, since the data contained three
cohorts, and the maximum number of sires and dams as 221
in each category, based on our knowledge of the size of the
breeding population. However, no pedigree information was
included in this model.
Since the relatedness matrix generated by simulated
annealing is guaranteed to be positive definite, this presented
the opportunity to explore how error in the estimation of
relatedness affected the performance of the animal model.
One approach to reducing the error in R is to reduce the
dimensionality (or rank) of this symmetrical matrix. In such
a high-dimensional space, the vast majority of independent
dimensions will represent very minor axes of variance that
are likely to reflect measurement error in individual
procedure that enabled an investigation of which subspaces
of the relatedness space represented by R were more closely
associated with phenotypic similarity. Our approach is
similar to that implemented in investigations of the
dimensionality of identity-by-descent matrices in the context
of variance-component quantitative trait locus analysis
(Ro ¨nnega ˚rd & Carlborg 2007; Ro ¨nnega ˚rd et al. 2007).
Pedigree-free animal models
F. D. Frentiu et al.
Proc. R. Soc. B (2008)
A spectral decomposition of R is given by
where U is an upper triangular matrix of coefficients
representing the eigenvectors of R; L is a diagonal matrix
containing the eigenvalues of R; and superscript Trepresents
the transpose of a matrix. A reduced-rank R (Rn-d) where d
is the number of dimensions may then be obtained by
defining a subspace of R by selecting a subset of columns of
U (Un-d) and applying
Rn-dcan now be substituted for R in the animal model. All
matrix manipulations were conducted in SAS IML
We present the results of analyses that used four reduced-
and R13-dand R6-drepresenting 50, 32, 27 and 23% of the
estimated variance in relatedness, respectively (appendix 2 in
the electronic supplementary material). The first two of these
reduced-rank matrices were chosen for the level of variance in
an examination of the scree plot of eigenvalues of R (see §3).
We implemented the animal model in this case using the
MIXED procedure in SAS (2001). As in the traditional animal
model, analyses were based on nestling data and included
year of sampling, hatching date and their interaction as fixed
effects, and nest of rearing as an additional random effect.
Univariate animal models were run for all traits to estimate
genetic variances, and genetic covariances were estimated by
first estimating the genetic variance in the sum of a particular
bivariate trait combination, then taking away the genetic
variance of each trait and dividing by two (Mezey & Houle
2005). Substantial problems were encountered in the
convergence of REML-based models when the R matrix
was substituted into the animal model, with many parameters
unable to be estimated in initial analyses. We subsequently
determined that three individuals needed to be excluded from
the dataset to allow convergence: two birds based on outlying
phenotypic values and one based on the convergence
diagnostics provided by the MIXED procedure in SAS (2001).
(iv) Comparison of G matrices
Finally, we compared the genetic variance–covariance (G)
matrices derived from each of the three methodologies. Since
the three methods were not all implemented in the same
model, log-likelihood ratio tests for testing differences among
genetic parameters in the matrices were not available to us.
We therefore adopted a geometric approach that compared
the similarity of three-dimensional subspaces of the G
where the matrices A and B contain the first three
eigenvectors of the two G matrices to be compared as
columns (Blows et al. 2004). The sum of the eigenvalues of S
is then a bounded measure of the similarity of the two
subspaces (between 0 and 3, with 3 indicating coincident
subspaces; Blows et al. 2004).
(a) Molecular markers and genotyping
In total, 44 alleleswerefound at 11 polymorphic loci in the
silvereye population (appendix 1 in the electronic
supplementary material). Although allelic diversity was
moderate, with a maximum of seven alleles at the most
polymorphic locus, observed heterozygosities were high
and did not deviate from those expected (appendix 1 in the
electronic supplementary material). When we calculated
exclusion probabilities for parentage assignment we found
that, although each locus individually had a low exclu-
sionary power, the exclusion power of the full set
(b) Marker-based estimates of relatedness
In general, all three of the widely used algorithms for
estimating pairwise relatedness performed well in differ-
entiating between known relatives and unrelated individ-
uals (figure 1), with the mean estimated relatedness for full
siblings being consistently significantly higher than
that for ‘unrelated’ pairs of individuals (QR: tZ15.17,
d.f.Z58, p!0.001; LR: tZ13.28, d.f.Z58, p!0.001;
W: tZ11.95, d.f.Z58, p!0.001). The simulated
annealing approach to estimating relatedness also success-
fully differentiated between known relatives and unrelated
individuals (figure 1), with mean estimated relatedness
for full siblings being significantly higher than for
putatively unrelated pairs of individuals (FT: tZ8.86,
d.f.Z58, p!0.001). However, this approach tended to
underestimate the absolute level of relatedness of
full siblings, for which it estimated an average relatedness
We were unable to detect significant variance in
relatedness using any of the traditional estimators, with
var(rij) ranging from K0.001 to 0.008 and all standard
errors being large (QR: var(rij)Z0.005 (G0.012), LR:
var(rij) K0.001 (G0.005) and W: var(rij) Z0.008
(G0.011)). Despite the dataset consisting of full-sib
families, the proportion of all pairwise comparisons that
were among full-sib pairs was approximately 1%.
QR LRW FT
mean relatedness ± 2 × s.e.
Figure 1. Performance of four different algorithms in
estimating genetic relatedness between individuals: QR
(Queller & Goodnight 1989), LR (Lynch & Ritland 1999),
W (Wang 2002) and FT (Ferna ´ndez & Toro 2006). Mean
relatedness denoted by circles (full-sibling pairs) or triangles
(putatively unrelated pairs). Bars indicate twice the standard
error. Full-sibling pairs were inferred from behavioural
observations and COLONY (Wang 2004a,b). Samples consist
of 30 independent pairs of either full siblings or putatively
unrelated individuals and are identical for each estimator.
642F. D. Frentiu et al.Pedigree-free animal models
Proc. R. Soc. B (2008)
(c) Estimating quantitative genetic parameters
(i) Traditional animal model
The results of the traditional animal model analysis based
on known pedigree information were reported in detail in
Frentiu et al. (2007). Briefly, this method revealed
substantial additive genetic variance for all six morpho-
logical traits, with genetic variances ranging from 0.13 to
0.24 (table 1a). There was also positive genetic covariance
between all six morphological traits (range 0.06–0.22;
table 1a). Finally, this method also revealed significant
common environmental effects (nest effects) for four of the
six traits, the two non-significant traits being culmen width
and culmen depth (range 0.03–0.24; table 2).
(ii) Ritland pairwise regression method
Spectral decomposition showed that pairwise relatedness
estimated using the QR estimator results in anon-positive-
definite relatedness matrix, with approximately half of the
eigenvalues being negative (figure 2a).
Ritland’s pairwise regression method provided esti-
mates of genetic variance (range K0.001 to 0.002) and
covariance (range K0.001 to 0.002) that were very much
smaller than those derived from the traditional animal
model, with many of the estimates being effectively zero or
negative (table 1b). Note that these results supersede those
previously reported for this population (Frentiu 2004),
and reviewed elsewhere (Garant & Kruuk 2005), which
were based on a subset of the data used here.
(iii) Molecular genealogy method
Spectral decomposition confirmed that the simulated
annealing algorithm successfully returned an R matrix
that was positive definite (i.e. the eigenvalues were positive
for all eigenvectors; figure 2b). The scree plot in figure 2b
suggested that there may have been a qualitative (although
small) difference in the amount of variance explained by
the eigenvectors after eigenvectors 6 and 13. We therefore
included the R6-d and R13-d reduced-rank matrices in
animal models, in addition to two further matrices, R25-d
and R90-d that explored the effect of many more
dimensions in R on the estimation of genetic variance,
but still removed 68 and 50% of the estimated variance in
relatedness from the analysis, respectively.
When we used the full R matrix to estimate genetic
parameters, we found that only three of the six traits had
estimates of genetic variance that were non-zero
(appendix 2 in the electronic supplementary material).
Subsequent tests indicated that the R25-dand R13-dwere
the best performing of the models. The R13-danalysis
returned all non-zero genetic variances, while the R25-d
analysis returned only three non-zero genetic variances
and the R25-danalysis returned smaller genetic variances
than the R13-danalysis for each trait.
We analysed the data using the R13-din two further
respects. First, we tested for a common environment
Table 1. Additive genetic variance–covariance matrices derived from three different methodologies: (a) traditional pedigree-
basedanimal model, (b) Ritland’s pairwise regression, and (c) pedigree-free animal modelbased on R13-d, the 13-dreduced-rank
molecular genealogy. (Genetic variances are shown in bold on the diagonal with genetic covariances below the diagonal.)
wing lengthtail length tarsus lengthculmen lengthculmen depthculmen width
(a) traditional pedigree-based animal model (from Frentiu et al. 2007)
(b) Ritland pairwise regression method
tarsus length 0.000
(c) pedigree-free animal model based on R13-d, the 13-d reduced-rank molecular genealogy
Table 2. Estimates of common environmental effects (nest
effects) from the traditional animal model and the pedigree-
free animal based on R13-d. (Significance is indicated by
type of model
Pedigree-free animal models
F. D. Frentiu et al.
Proc. R. Soc. B (2008)
variance component (nest effect) for each trait in turn
(table 2). The reduced-rank R matrix indicated signi-
ficant common environment effects for five of the six
traits (range 0.016–0.304), with the non-significant trait
being culmen width, which is also one of the two traits
that were non-significant in the traditional animal model
(table 2). Second, we estimated the genetic covariance
among traits that were found to be very different from
those obtained using the traditional animal model, all
being negative (table 1c). It is worth noting, therefore,
that the overall G matrix estimated by the pedigree-free
method, shown in table 1c, breaks the rule that
variance–covariance matrices must be positive definite
because it contains estimates of covariance that are
negative. This is presumably because we estimated each
component separately, rather than in a single multi-
variate analysis, and the covariance estimates themselves
are not very robust.
(iv) Comparison of G matrices
The comparison of the three-dimensional subspace
between G from the traditional animal model and
Ritland’s regression method gave a value of only 1.20 for
the sum of the eigenvalues of S, indicating a low level of
similarity between the two G matrices. The comparison
between the traditional animal model and the pedigree-
free animal model based on R13-dgave a value of 1.62,
indicating a marginally better performance in recovering
the genetic covariance structure found by the traditional
Our results offer a mixed assessment of the potential use of
pedigree-free animal models in estimating quantitative
genetic parameters in natural populations. On one hand,
we have shown that it is possible to implement animal
models using a genealogical matrix based exclusively on
molecular marker data. In addition, the estimates of
genetic variance obtained using this method were of a
similar overall magnitude to those obtained using a
traditional animal model. Reducing the dimensionality
of the molecular R matrix increased the ability of the
pedigree-free animal approach in recovering genetic
variance, presumably by improving the signal to noise
ratio. On the other hand, the estimates of covariance
obtained using the pedigree-free animal approach tended
to be negative, whereas the traditional animal model
showed positive covariance among most traits, and the
overall similarity in G matrices between the two tech-
niques was only moderate. Taken together, these results
indicate that, while pedigree-free animal models can
indeed recover quantitative genetic information, the full
pattern of covariance might be difficult to detect.
Our results were less encouraging regarding the
potential use of Ritland’s pairwise regression method, at
least for this population. The pairwise regression method
also commonly yielded negative estimates of heritability,
additive genetic variance and additive genetic covariance.
In addition, the overall structure of the genetic variance–
regression technique was more different from that
obtained using the traditional animal model method
than that recovered from the molecular genealogy
approach. The relatively poor performance of the
pairwise method should, of course, be assessed cautiously
because both the other types of model included
additional explanatory factors (year of measurement,
hatching date and the interaction between these two
effects), which may have improved the performance of
those models. On the other hand, we feel that the
difficulty of including such factors in pairwise regression
models is an inherent limitation to the development of
this approach, along with the fact that the pairwise
relatedness estimates cannot be rearranged in matrix
form and used in an animal model, as such matrices are
negative definite. Thus, in combination with the difficulty
of estimating statistical significance using the pairwise
regression method, and the fact that several other tests
based on this approach have also reported problems in
obtaining reliable estimates (Klaper et al. 2001; Thomas
et al. 2002; Wilson et al. 2003; Andrew et al. 2005;
Coltman 2005; Shikano 2005; van Kleunen & Ritland
2005), we agree with previous reviews that the prospect
of applying this method to a wide range of populations
may be limited (Garant & Kruuk 2005; Owens 2006).
Why did the pedigree-free approach fail to recover the
full pattern of quantitative-genetic covariance? The core
challenge in studies of this type is lack of significant
variance in relatedness among individuals in the popu-
lations according to estimates based on molecular marker
data (Ritland 1996; Csille ´ry et al. 2006). There are,
however, two different types of explanation for low
variance in relatedness, which have very different impli-
cations for the prospects of the pedigree-free approach.
The first explanation for low relatedness variance is that
rank number of eigenvectors
50 100 150 200 250 300 350 400 450 500
Figure 2. Scree diagrams of eigenvectors for (a) the
relatedness matrix used in the Ritland pairwise regression
analyses and (b) the relatedness matrix used for the pedigree-
free animal model based on the molecular genealogy. The
arrow shows the point along the eigenvector axis after which
eigenvalues become consistently negative.
644F. D. Frentiu et al.Pedigree-free animal models
Proc. R. Soc. B (2008)
this is an intrinsic property of the population. Under this
scenario, even an improved set of molecular markers would
not increase relatedness variance because such variation is
simply not present (Csille ´ry et al. 2006). In our particular
variance may be due to the island-dwelling nature of the
population. Indeed, it has recently been suggested that low
relatedness variance may be a general property of wild
populations (Csille ´ry et al. 2006), and therefore that the
pedigree-free animal model may always struggle to obtain
sufficient power to estimate complex genetic patterns.
However, our finding that reducing the dimensionality of
the molecular G matrix increased our ability to detect
genetic variance argues against this possibility.
The second explanation for the apparent inability to
recover the level of heritability found by the traditional
animal model is that sufficient variation in relatedness
does exist in the population, but we were unable to detect
that variation (Csille ´ry et al. 2006). This could be because
either the batteryof molecular markers employed here was
not sufficiently powerful to reveal that variation or the
variation is obscured through limitation in the algorithm
used to estimate the genealogy. There is some evidence to
support the influence of both these sources of error
variance. For instance, the molecular markers used have
relatively low polymorphism so, although we have shown
they are sufficiently powerful to potentially identify first-
degree relatives, they may fail to differentiate among more
distant relatives. Similarly, it is clear from the absolute size
of the relatedness estimates in figure 1 that, although the
algorithm used to construct the molecular genealogy
(Ferna ´ndez & Toro 2006) can, on average, distinguish
successfully between close genetic relatives and unrelated
pairs of individuals, the same algorithm tends to under-
estimate the genetic relatedness of full sibs. This under-
estimation is likely to be caused by a combination of a
weak signal due to the low polymorphism of the molecular
markers and the search procedure of the algorithm itself.
Improving the performance of such methods is clearly a
priority for future work if molecular genealogies are to be
widely applied in animal models.
It remains an open question, however, whether natural
populations in general will have sufficient variance in
relatedness to facilitate the pedigree-free animal model
approach or whether our study population is unusual in
this respect. Our population was an isolated island
population with low to moderate levels of genetic
polymorphism, and our experimental design ensured
that we had a sample richer in full siblings than one
might expect at random. Nevertheless, iffuture studies do
reveal substantial variation in relatedness in other wild
populations, then our analyses suggest that the pedigree-
free animal model may provide an opportunity to study
the evolutionary genetics of wild populations, particularly
if its use is complemented by the development of
increasingly powerful molecular markers, more sophis-
ticated analytical methods for reconstructing genealogies
from marker data, and the subsequent dimension
reduction of the resulting relatedness matrix.
All work was conducted under permits obtained from The
Australian Bird and Bat Banding Scheme, the Environment
Protection Agency of Queensland and the Animal Ethics
Committee of the University of Queensland.
We thank the Heron Island Research Station, P & O Heron
Island Resorts, Queensland Marine Parks, NERC Molecular
Genetics Facility (Sheffield), S. Chenoweth, D. Dawson,
C. Edwards, J. Hadfield, K. Henderson, W. Hill, J. Kikkawa,
L. Kruuk, A. Krupa, M. Losiak, B. Maher, F. Manson,
K. Ritland, S. Robinson-Wolrath, S. Scott, B. Sheldon,
S. Twigg, B. Walsh, J. Wang and C. Wiley for help and
discussion. This work was supported through funding from
the Australian Research Council and the Natural Environ-
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